0,8

Terms for n <= 30 come from Table 1 of the Gallot et al. paper, which quotes results from Moller. Sequence A138475 gives the minimum order of the cyclotomic polynomial that produces that maximal coefficient. A very fast method (due to Grytczuk and Tropak) for computing the coefficients up to x^k in the cyclotomic polynomial Phi(n,x) is given by the Mathematica function coef[k,n] below.

The first n for which a(n)>n is 118. The sequence appears to be monotonic for n>143. Terms up to n=128 were found by exhaustive search; subsequent terms were found by a much faster hill-climbing method.

A. Grytczuk and B. Tropak, A numerical method for the determination of the cyclotomic polynomial coefficients, Computational number theory (Debrecen, 1989), 15-19, de Gruyter, Berlin, 1991.

H. Moller, Uber die i-ten Koeffizienten der Kreisteilungspolynome, Math. Ann. 188 (1970), 26-38.

T. D. Noe, Table of n, a(n) for n=0..1000

Yves Gallot, Pieter Moree and Huib Hommersom, Value distribution of cyclotomic polynomial coefficients

a(7)=2 is attained for the cyclotomic polynomial Phi(105,x), which has the term -2x^7.

coef[k_, n_] := Module[{t, b=Table[0, {k+1}]}, t=-MoebiusMu[n]*Table[g=GCD[n, k-m]; MoebiusMu[g]*EulerPhi[g], {m, 0, k-1}]; b[[1]]=1; Do[b[[j+1]] = Take[b, j].Take[t, -j]/j, {j, k}]; b]; Table[mx=1; r=PrimePi[k]+1; mnN=Prime[r]; ps=Reverse[Prime[Range[r]]]; Do[d=IntegerDigits[i, 2, r]; n=Times@@Pick[ps, d, 1]; c=Abs[coef[k, n][[ -1]]]; If[c==mx, mnN=Min[mnN, n], If[c>mx, mx=c; mnN=n]], {i, 2^r-1}]; mx, {k, 2, 20}]

Cf. A013594 (smallest order of cyclotomic polynomial containing n or -n as a coefficient).

Sequence in context: A093320 A082370 A005136 this_sequence A058761 A050119 A097637

Adjacent sequences: A138471 A138472 A138473 this_sequence A138475 A138476 A138477

nonn

T. D. Noe (noe(AT)sspectra.com), Mar 19 2008, Apr 14 2008, Feb 16 2009